Book cover for Calculus

Calculus

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

ISBN #9781119379331

7th Edition

5,101 Questions

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33,688 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section explores how to measure speed by computing average velocity over a time interval and progressing to the concept of instantaneous velocity using limits. The average velocity provides a general idea of movement, while the instantaneous velocity, derived as a limit, gives the exact rate of positional change at a specific moment. The graphical interpretation using slopes demonstrates that zooming in on a curve yields a line whose slope represents the instantaneous rate of change, forming the basis of the derivative concept in calculus.

Learning Objectives

1

Compute the average velocity of an object over a given time interval.

2

Define and calculate instantaneous velocity using the limit of average velocities as the interval shrinks.

3

Interpret the significance of the sign (positive or negative) of velocity in physical contexts.

4

Relate the concept of slope to the graphical representation of velocity and understand how zooming in on a curve leads to a tangent line.

5

Apply algebraic techniques to simplify difference quotients and determine exact limits.

Key Concepts

CONCEPT

DEFINITION

Average Velocity

The change in position divided by the change in time over an interval; represented as (s(b) - s(a))/(b - a).

Instantaneous Velocity

The velocity of an object at a specific moment, defined as the limit of the average velocity as the interval approaches zero: lim(h→0) [s(a+h) − s(a)]/h.

Derivative

A function that gives the instantaneous rate of change (or slope) of a function at any given point; in this context, it represents instantaneous velocity.

Limit

A fundamental concept in calculus that describes the value a function approaches as the input approaches some value.

Slope of a Curve

At a very small scale, the curve appears linear and its slope represents the instantaneous rate of change, or velocity, at that point.

Example Problems

Example 1

The distance, $s,$ a car has traveled on a trip is shown in the table as a function of the time, $t,$ since the trip started. Find the average velocity between $t=2$ and $t=5$ $$\begin{array}{c|c|c|c|c|c|c}\hline t \text { (hours) } & 0 & 1 & 2 & 3 & 4 & 5 \\\hline s(\mathrm{km}) & 0 & 45 & 135 & 220 & 300 & 400 \\\hline\end{array}$$

Example 2

The table gives the position of a particle moving along the $x$ -axis as a function of time in seconds, where $x$ is in meters. What is the average velocity of the particle from $t=0$ to $t=4 ?$ $$\begin{array}{c|c|c|c|c|c}\hline t & 0 & 2 & 4 & 6 & 8 \\\hline x(t) & -2 & 4 & -6 & -18 & -14 \\\hline\end{array}$$

Example 3

The table gives the position of a particle moving along the $x$ -axis as a function of time in seconds, where $x$ is in angstroms. What is the average velocity of the particle from $t=2$ to $t=8 ?$ $$\begin{array}{c|c|c|c|c|c} \hline t & 0 & 2 & 4 & 6 & 8 \\\hline x(t) & 0 & 14 & -6 & -18 & -4 \\\hline\end{array}$$

Example 4

Figure 2.6 shows a particle's distance from a point as a function of time, $t .$ What is the particle's average velocity from $t=0$ to $t=3 ?$ GRAPHS CANT COPY

Example 5

Figure 2.7 shows a particle's distance from a point as a function of time, $t .$ What is the particle's average velocity from $t=1$ to $t=3 ?$
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Step-by-Step Explanations

QUESTION

Compute the average velocity of the grapefruit over the interval 4 ≤ t ≤ 5 when its height changes from 150 ft to 106 ft.

STEP-BY-STEP ANSWER:

Step 1: Determine the change in height: 106 - 150 = -44 feet.
Step 2: Compute the time interval: 5 - 4 = 1 second.
Step 3: Divide the change in height by the time interval: -44/1 = -44 ft/sec.
Step 4: Interpret the result: The negative value indicates that the grapefruit is moving downward.
Final Answer:

Calculating Average Velocity

QUESTION

Determine the instantaneous velocity of a particle whose position function is s(t) = t² at t = 3 using the limit process.

STEP-BY-STEP ANSWER:

Step 1: Write the difference quotient: (s(3+h) - s(3)) / h, where s(t)=t².
Step 2: Substitute: ((3+h)² - 9) / h.
Step 3: Expand the numerator: 9 + 6h + h² - 9 = 6h + h².
Step 4: Factor h from the numerator: h(6 + h).
Step 5: Cancel h (assuming h ≠ 0): 6 + h.
Step 6: Take the limit as h → 0: lim (h→0)(6 + h) = 6.
Final Answer: The instantaneous velocity at t = 3 is 6 ft/sec.

Calculating Instantaneous Velocity using Limits

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Common Mistakes

  • Confusing average velocity with instantaneous velocity: Averaging data over an interval does not yield the exact speed at any specific time.
  • Ignoring the significance of negative velocity values, which indicate direction (e.g., moving downward).
  • Failing to cancel common factors in the difference quotient when simplifying limits.
  • Misinterpreting a graph’s slope by not properly associating it with instantaneous rather than average velocity.